How do you calculate the exact pixel dimensions when resizing an image from one aspect ratio to another?

An aspect ratio is the reduced fraction of width to height, and a resize preserves quality only when both dimensions move by the same factor. Enter your current pixel width and height, choose the target ratio, and this calculator returns the reduced ratio, the exact decimal multiplier, the megapixel count, the diagonal in pixels and the largest clean 300 DPI print, together with the pixel width or height you would need to hit the target shape without stretching anything.

What is the mathematical formula for image scale factor preservation without stretching the picture?

Aspect multiplier  = width / height
Reduced ratio      = (width / GCD) : (height / GCD)
Proportional scale = new width / old width = new height / old height
Height from width  = width / target multiplier
Width from height  = height x target multiplier
Megapixels         = width x height / 1,000,000
Diagonal (px)      = sqrt(width^2 + height^2)
Print size (in)    = width / 300  x  height / 300
Guardrails         = pixels clamp(10, max), ratio terms clamp(1, 32)

What is the full reference chart of standard video and photo resolutions from SD up to 8K?

Why does the same 16:9 shape appear at 1280x720, 1920x1080 and 3840x2160 without any visual difference?

Because the ratio is a shape and the resolution is a density. All three reduce to 16:9 once divided by their greatest common divisor, so they fill an identical screen area with no bars. What changes is the number of samples inside that shape: 0.92 MP, 2.07 MP and 8.29 MP respectively. More samples survive cropping, scaling and compression better, which is why editors shoot 4K even when the final delivery is 1080p.

How large can you print a digital photograph before the individual pixels start to become visible?

Divide each pixel dimension by 300 to get inches at photographic print quality. A 1920 × 1080 frame therefore prints at 6.40 × 3.60 inches, and a 24 MP camera file at 6000 × 4000 prints at 20 × 13.33 inches. Posters and billboards routinely drop to 150 or even 72 DPI because the viewing distance grows with the print, which doubles or quadruples the usable size from the same file.